Nanoscale sensing based on nitrogen vacancy centers in single crystal diamond and nanodiamonds: achievements and challenges

The major factors influencing diamond's properties and its potential applications are purity and crystallinity. The following retrospective summarizes recent developments within diamond growth and synthesis according to those parameters. In theory, diamond is a crystal consisting only of sp³ hybridized covalently bound carbon atoms, which build a repetitive unit cell made of fused hexagonal chair cyclohexane conformations as depicted in figure 1(a) [16]. The Hermann Mauguin notation classifies diamond into the Fd3m-07h face centered cubic group. The unique arrangement of carbon atoms within diamond makes it not only the hardest material in Mohs scale, but also causes it to be inert towards chemical modifications without the use of harsh conditions. Diamond's wide bandgap (indirect gap 5.45 eV) results in optical transparency for light up to ≈230 nm. The naturally almost spin free lattice of diamond (¹²C, 98.9%, I = 0, [7]) has been engineered by isotopically purifying diamond (residual ¹³C, 0.3%) as presented in [7].

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There are many types of diamonds and their classification is based on several factors. One classification system has its foundation in the origin of the diamonds: naturally occurring diamonds, diamond created via synthetic hydrothermal high-pressure high temperature (HPHT) protocols, shock wave synthesis (e.g. through detonation) and low pressure pyrolysis of hydrocarbon gases in chemical vapor deposition (CVD). The latter includes microwave assisted CVD (MWCVD) and hot filament CVD deposition, whereas MWCVD is most suitable to grow high purity diamonds [19]. Figure 2 summarizes this classification. For CVD diamond growth, the most common precursor gases are methane CH₄ and hydrogen H₂. The detonation method only forms nanodiamonds (NDs) of about 5 nm size [20]. Figure 1(b) summarizes the temperature/pressure regimes for diamond synthesis through shock wave, metastable CVD diamond phase generation or HPHT protocols. In both HPHT and CVD synthesis, by tailoring the additives in the reaction chamber (e.g. insertion of N₂), diamonds with various color centers are obtained. For detonation diamonds (DNDs), creating luminescent NDs is challenging. However, recently density control of NV centers in 5 nm-sized DNDs was reported [21]. Usually DNDs require special post-treatment to remove graphitic sp² layers, e.g. through immersion in a boiling mixture of oxidizing mineral acids (HNO₃, H₂SO₄ and HClO₄) and thermal annealing in vacuum or under oxygen atmosphere for adequate stabilization of spectral properties [22]. Various synthesis methods generate diamond with tailored properties. For example DNDs are commonly used as abrasives in industry, while HPHT and CVD (especially single crystal) diamonds are often more suitable for quantum applications.

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For HPHT and CVD methods, diamond growth is typically initiated on a substrate or seed. For CVD growth, this can be attained by directly growing on a diamond (homoepitaxy) or via using a non-diamond substrate (heteroepitaxy) which needs to be prepared suitably. Substrate pretreatments for CVD span from seeding with NDs to grow polycrystalline diamond to creation of nuclei using bombardment with carbon ions (bias enhanced nucleation for heteroepitaxy) [23]. Homoepitaxy limits the size of grown single crystal diamonds (SCDs) due to the size of available substrates. This can be overcome using heteroepitaxy [23].

The smallest possible seeds for diamond growth are so called diamondoids. Diamondoids mainly combine one or more adamantane molecules (C₁₀H₁₆), the smallest unit cage structure of the diamond crystal lattice. Using the smallest possible seed is among other reasons motivated by minimizing the influence of the seed on the resulting ND's properties. References [16, 24] demonstrate the synthesis of nanoscale 'diamond molecules' (higher diamondoids, e.g. tetramantanes, less than 1 nm in size). Diamondoids were recently employed to enable the growth of high quality NDs in a HPHT process [25] and in a CVD process [26]. Reference [27] reports that diamondoids serve as nuclei in CVD if they contain more than 26 carbon atoms. Another approach to molecular sized diamonds is the use of meteoritic diamonds [28]. For this material class, fluorescent SiV centers have been observed in NDs with only 1.6 nm size. However, this result has never been obtained using man-made NDs, thus significantly lowering applicability. A novel route to manufacturing an extreme diamond nanomaterial is the recently observed transformation of a bilayer of graphene into a single layer of (fluorine terminated) diamond [29].

The second system that classifies diamond, in addition to the origin of the diamond, is based on crystallinity. Here diamonds are classified as SCD, polycrystalline and (ultra-)nanoscrystalline diamonds according to their crystallite size. For grain sizes below 10 nm the diamond is typically called ultrananocrystalline, between 10 and 50 nm nanocrystalline, and from 50 nm to 500 μm micro- or polycrystalline. If the crystallite size exceeds 500 μm, the diamond is termed single crystalline and a specified crystallographic growth plane can be assigned [30].

The amount of publications detailing the phrase 'SCD growth' in WorldCat.org in the time period between 2013 and 2018 yields 3.575 results (figure 3), illustrating the significance of the field.

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Free standing wafer-sized polycrystalline diamond films are currently commercially available with diameter up to 140 mm (Element Six, UK). Recently [23], presented a mm-thick SCD wafer with 98 mm diameter grown by heteroepitaxy. The basic usability of this material for sensing with NV centers has been demonstrated [31].

The most common growth direction in CVD is (100). However, for many applications it is advantageous to control the orientation of NV centers with respect to the SCD surface. Thus, alternative growth of (111) [32] diamond and (113) [33] has been realized that orients the dipoles of NV centers more optimal with respect to photonic structures manufactured into the SCD surface [34]. While SCD allows for the highest purity material with the lowest background PL, poly- and nanocrystalline forms will always suffer from broadband PL occurring at grain boundaries [19].

The third classification has been historically established for natural diamonds and classifies them according to their optical absorption [35]. Figure 2 summarizes this classification. Type IIa diamonds were classified by a lack of optical absorption [35], so this were the purest diamonds in this classifications. The absorption of type I diamond is nowadays assigned to nitrogen [36]. Type Ia diamonds contain nitrogen in aggregated form; while in type Ib diamonds, it is present in its isolated form substituting single carbon atoms (substitutional nitrogen) [36]. Most CVD diamonds are of IIa type, while HPHT diamonds are often type Ib [37]. In contrast, type I diamonds are known to occur most commonly in nature. Type IIb contains boron in small quantities and has p-doped character. This historical classification does not sufficiently classify diamonds typically used for sensing with color centers: these diamonds are mostly so pure that they are all class IIa but need further classification. So typically, the amount of substitutional nitrogen and boron is directly given. Pushing purity to the limit, electronic grade diamonds with less than 5 ppb nitrogen and less than 1 ppb boron are nowadays commercially available from ElementSix and are the basis for most sensing experiment especially using individual NV centers. In addition, ¹²C isotopically-purified diamond has been demonstrated and termed quantum grade diamond [7].

When impurities in diamond are present in high concentration, they induce a coloration of diamonds. For sensing, typically low concentrations of defects are used, which does not necessarily lead to a coloration of diamond.

Color centers can be introduced into diamond using two approaches. The first approach creates color centers in the growing diamond via using selected precursors [38–40]. Often NV centers are also created due to background nitrogen in the process [34, 40–42]. However, for sensing applications it is important to carefully control color center density and depth below the diamond surface (see also section 4). Creating thin layers containing NV centers has been attained in diamond CVD using δ-doping (e.g. [38, 43]): in a CVD process with low growth rate, nitrogen gas is introduced for a short time thus inducing an only 6 nm thick nitrogen-doped layer [38]. Moreover, nitrogen-doped layers were obtained by overgrowing nitrogen-terminated diamond surfaces [44]. Using δ-doping, conversion of nitrogen impurities to NV centers is often fostered via irradiation e.g. using helium ions [45]. One advantage of forming color centers directly during diamond growth is that they can show preferential alignment: instead of aligning along all equivalent directions in the crystal, color centers only form with one orientation leading to a perfectly aligned ensemble of centers (see [39–41]). Moreover, NV centers created using conversion of grown-in nitrogen to NV centers have been found to have superior properties compared to centers arising due to implanted nitrogen [46, 47].

The second approach creates impurities after diamond manufacturing has finished. It utilizes irradiation (by electrons, neutrons or protons [46, 48]) and ion implantation [49–54]. It allows to control the fluence, which determines the density and the energy (ranging from keV to MeV) to set the implantation depth [55]. The creation of color centers can be also controlled in lateral dimension by implanting through a pierced AFM tip [56], using nanomasks [57] or focused ion beams [58]. Ion implantation in diamond requires post-treatment annealing (mostly at temperatures exceeding 600 °C) in order to induce vacancy migration, repair implantation damage and form color center complexes like NV centers.

To efficiently extract color center PL from SCD, photonic structures e.g. nanopillar waveguides [34, 59], optical antennas [60], solid immersion lenses [61], bulls eye gratings [62] or pyramids [63–65] are highly-desirable (for reviews on diamond photonics, see [66–69]). Figure 4 displays some examples of diamond nanostructures. Nanopillar waveguides (figure 4(b)) channel the light into conveniently collectible emission angles (NA < 0.95) and circumvent total internal reflection [70]. Here, the high refractive index of diamond (2.4) aids in confining the light in the waveguide. In contrast, for unstructured diamond, it limits the collection efficiency to only a few percent [71]. While optical antennas, in principle, allow up to unity collection efficiency [60], tip-like structures as pillars or pyramids (figure 4(c)) might simultaneously serve as a tip for a scanning probe microscope [72] thus enabling to scan a color center in close proximity to a sample under investigation. It is noteworthy that nanopillars have been produced incorporating not only NV centers [34, 70, 73] but also SiV centers [59] and lead-related defects [74]. Additionally, cavity structures like one or two-dimensional photonic crystals (figure 4(d)) [56, 75] or microdisk (whispering gallery mode) resonators (figure 4(b)) [76] allow the coupling of color centers to cavity modes with high Q-factor. Mostly, photonic structures are used to extract light from single color centers [59]. However, very recent studies also indicate the usefulness of pillar arrays for sensing with NV ensembles [73].

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To harness the full potential of photonic structures, nanofabrication processes must be carefully controlled: first, the shape of the nanostructure will strongly influence its performance e.g. changing the taper angle of a nanopillar (see figure 4(a)) changes the efficiency of light extraction [71, 77]. Second, nanofabrication, especially plasma etching, might damage diamond surfaces [78]. This damage was estimated to reach several nanometers into the diamond and found to deteriorate the properties of shallow color centers [79]. Reduced damage and enhanced properties of NV centers have been reported by using plasmas with zero bias voltage and thus minimal acceleration of ions in the plasma [79] and with careful combinations of etching processes [46]. As diamond is highly chemically inert, wet chemical etching methods are not an alternative to plasma-based processing. Focused ion beam milling leaves gallium impurities in the diamond that need to be removed using sophisticated post-processing [80]. So the most common processing techniques utilize standard nano-lithography techniques with electron beams, laser or 2-photon lithography for mask generation and transfer the patterns into diamond using reactive ion etching (RIE), mostly in the form of inductively coupled reactive ion etching (ICP-RIE) [81, 82]. Plasma-etching of diamond requires special gases as well as resists for the masks (negative and positive tone). Most commonly oxygen/argon or chlorine-based mixtures are used. Masks are either based on PMMA with hard mask techniques (Al, Al₂O₃, W) or directly on the negative tone resist hydrogensilsequioxane [34, 83].

Nanostructuring of diamond can be classified into traditional nanotechnology approaches: bottom-up and top-down approaches. For example nanodisc resonators with quality factor exceeding 300000 may be generated by top-down techniques involving electron-beam lithography, laser lithography and ICP-RIE [76].

In bottom-up approaches, nanostructures are directly formed during diamond growth thus potentially forming many structures at the same time and avoiding potential damage due to plasma-etching [64, 84]. However, top-down approaches have the advantage, that color centers can be created while the diamond is still in its bulk form, which eases annealing and cleaning followed by fabrication of the nanostructures (e.g. [72]).

The NV center consists of a nitrogen atom that substitutes a carbon atom in the diamond lattice and a neighboring vacancy. The level structure of the NV center is summarized in figure 5 and discussed in the followin8. Negatively-charged NV centers display a system of singlet and triplet levels. A strong optical transition (lifetime 12.9 ns, [87]) occurs between the ³A₂ ground state and the ³E excited state. This transition leads to the ZPL at 637 nm at room temperature. Due to phonon assisted transitions, the narrow ZPL is accompanied by a broad phonon sideband spanning almost 100 nm spectral width [88]. The ³A₂ → ³E transition, which induces the visible PL of the NV center, is represented by two orthogonal transition dipoles. These dipoles are placed in the plane orthogonal to the NV's high-symmetry axis that connects the nitrogen and the adjacent vacancy. This axis is oriented along one of the equivalent [111]-directions of the diamond lattice [89, 90].

In the triplet ground state of NV⁻, three spin states with m_s = −1, 0 and +1 are observed. Between the m_s = 0 level and the degenerate m_s = −1, +1 levels a zero-field splitting of 2.88 GHz occurs. These levels show spin-dependent PL: a NV center in m_s = 0 cycles between ³A₂ and ³E and produces bright PL. In contrast, a NV center in the m_s = −1 or +1 state undergoes an intersystem crossing to the singlet states with higher probability. As the center will reside in these states for typically around 200 ns [91], the visible PL of the center is reduced. The PL difference between the two spin states under continuous, non-resonant laser excitation (e.g. using 532 nm laser light) can be up to 20% [2]. Simultaneously, non-resonant laser excitation polarizes the NV center into the m_s = 0 state. Following this initialization, transitions between the spin levels can be driven using microwaves.

Between their ¹A₁ and ¹E levels (see figure 5), NV centers display a transition in the infrared spectral range (1042 nm) [88]. As a result of non-radiative processes, the PL due to this transition is typically four orders of magnitude weaker than the visible PL [92]. Nevertheless, transitions in the singlet system are a valuable resource for sensing (see section 3.2).

If one electron is removed from the NV⁻ center, it converts to a neutral NV⁰ center (ionization, see figure 5). This occurs during optical excitation of NV⁻ when a NV⁻ center which resides in the excited state absorbs an additional photon. Subsequently, an electron is excited to the conduction band and via an Auger process, a free electron is created, while the center converts to NV⁰ [93]. NV⁰ centers display a ZPL at 575 nm, thus, light with wavelengths shorter than 575 nm is needed to excite NV⁰. As a consequence, light with wavelengths between 637 and 575 nm only excites NV⁻ but not NV⁰. This can be used to detect the charge state of NV centers [42, 94] even without changing the charge state: only if the center is NV⁻, under excitation with e.g. 594 nm light PL occurs, while no PL is observed in the NV⁰ case. To convert NV⁰ back to NV⁻ it is necessary to excite NV⁰: If an electron from the lower orbitals of NV⁰ is excited, an electron from deep lying orbitals in the valence band can be promoted to the NV⁰ and re-charge the center to NV⁻ (see figure 5) [93]. It should be noted that the NV charge state in an ensemble can strongly depend on previous laser excitation of the NVs [95].

Sensing using NV centers can be classified into three broad categories, considering the physical quantity to be detected and the degree of freedom of the NV used:

• Detection of the (electronic) spin resonance frequencies (magnetometry, temperature sensing, conductivity measurement).
• Detection of NV PL intensity and excited state lifetime (near-field energy transfer, Förster-Resonance Energy Transfer (FRET) imaging, near-field sensing).
• Detection of the charge state of NV centers (sensing of electrostatic environment).

Measurements based on the spin degrees of freedom are most mature, while near-field based sensing and using the NV charge state for sensing are rather novel. Regarding electronic spin-based sensing, we introduce the sensitivity η for DC magnetometry [2]:

$$\begin{eqnarray}&&\eta =\displaystyle \frac{h}{g{\mu }_{B}}\displaystyle \frac{{\rm{\Delta }}\nu }{\sqrt{{I}_{0}}C},\end{eqnarray} \tag{ 1 }$$

where I₀ is the NV PL rate, C is the ODMR contrast, Δν the linewidth of the ODMR resonance, g the Landé factor and μ_B the Bohr magneton. Considering equation (1), improving sensitivity can be obtained by:

• Increasing I₀ via increasing the collected photon rates or the collection efficiency for NV PL [2, 34, 39, 40, 60, 62, 96–98].
• Decreasing Δν, which is connected to increasing the coherence time ${{T}}_{2}^{\star }$ [43, 44, 49–52, 79].
• Increasing the efficiency of the spin readout (in equation (1) represented via C) [50, 94, 99–105].

Equation (1) is valid for a single NV, if we use an ensemble including N NV centers, the sensitivity improves by a factor of $\sqrt{N}$. Indeed, to date, optimal sensitivities in the pT Hz^−1/2 are reached using ensembles of N ∼ 10¹¹ NVs [48]. For a recent review focusing on enhancing the sensitivity of NV ensemble-based sensors see [106].

For sensing schemes based on the NV charge state and FRET-based sensing and imaging, it is much more challenging to define a sensitivity. However, due to optical read-out, enhancing photon collection will enhance sensing capabilities. For FRET-based sensing, the quantum efficiency QE of the NV transitions, that is the probability for a radiative transition between excited and ground state, also plays a major role. Recent work indicates that the QE for shallow NVs, required for FRET-based sensing (4.5 nm below surface), amounts to 0.7 while defects deeper in the diamond reach QE of 0.86 [53].

In the following, we review the above mentioned methods and the main parameters influencing their sensitivity.

3.2.1. Collection efficiency

3.2.2. Coherence time

3.2.3. Readout and charge state

3.2.4. Ensemble density and alignment of NVs for wide field imaging

While NVs in high-purity SCD typically have superior properties compared to NVs in often less pure NDs, probing sample dynamics with high spatial resolution requires to transfer the sample under investigation onto the SCD surface and spatial resolution is typically limited by diffraction to around 0.5 μm. Nevertheless, shallow NVs in SCD have been used for various applications probing micron-scale dynamics especially in the context of novel materials and current imaging. The almost back-action-free detection of currents in graphene, where currents in the μAmpere regime are measured with a resolution of 0.5 μm [139], and the usage of NV centers to investigate failure of integrated circuits [125] illustrate their potential to analyze currents on the micron scale. Very recently, NV centers measured the band bending inside a diode like device in-situ [54]. Whereas in [140] shallow NV centers are used as sensor to perform nuclear magnetic resonance spectroscopy of atomically-thin hexagonal boron nitride layers. Thus NV centers allow to bring the technology of magnetic resonance truly to the nanoscale. For probing liquid samples, Kehayias et al structured SCD to form a microfluidic chip and flow the substance under investigation through the diamond device [141].

In contrast to the above mentioned applications which all rely on the spin-based sensing capability of NVs, sensing relying on the dipolar nature of the NV center is less advanced. There have been first demonstrations using near-field-based energy transfer (FRET) to detect dye molecules attached to the surface of NDs [142, 143]. Additionally, a scanning ND attached to an AFM tip has been used to image graphene flakes with nanoscale resolution [3]. Other authors reported non-successful attempts to establish FRET using NDs potentially due to a lack of control over the surface properties of NDs [144]. Consequently, it would be highly-advantageous to broaden FRET-based sensing to shallow color centers in SCD. For sensing and imaging approaches using the dipolar nature of NV centers, it is significant to consider the role of the two electric transition dipoles, $\vec{X}$ and $\vec{Y}$ as we discuss in detail in the appendix. As a first step toward FRET-based imaging using color centers in SCD, we here present quenching due to graphene deposited onto the SCD surface.

We use commercial, high-purity, CVD, electronic grade SCD from Element Six ([N]^s < 5 ppb, B < 1 ppb). To create shallow NVs, we implant the sample with 6 × 10¹¹ cm⁻² nitrogen ions (Innovion, USA). Via annealing to 800 °C in vacuum and cleaning in boiling acids (nitric, sulfuric, perchloric acid, 1:1:1), we create a layer of NV centers on average 7 nm below the SCD surface. From this clean surface, we observe homogeneous NV PL. We estimate the number of NV centers in the laser focus of our confocal microscope to be on the order of <10. The NV ensemble shows a spatially consistent lifetime of 15.5 ± 1 ns, which is slightly higher than the bulk lifetime of 12.9 ± 0.1 ns [87] due to a decrease in effective refractive index close to the surface. We now apply graphene flakes from solution (pristine graphene monolayer flakes from graphene supermarket with an average flake size of 550 nm (150–3000 nm), dispersed in ethanol). We spin coat this solution as-received with 600 rpm onto the SCD surface. We observe a slight trend to agglomeration of graphene. While in principle not desired, these agglomerates enable to straightforwardly observe the interaction of the NV centers with the graphene: we identify agglomerates via white light illumination of the SCD surface as well as Raman spectroscopy revealing a G Band at 1586 cm⁻¹. At the position of graphene agglomerates, we observe a reduced PL intensity (see figure 6(a)) as well as a clearly decreased NV lifetime (see figure 6(b)). We check our results on several positions on the sample and find typical results as shown in figures 6(a)–(c). From the decrease in lifetime and the graphene/NV Förster radius of 15 nm [3], we estimate the average distance of the graphene to the NV centers to be 19 nm which is higher than the average depth of the NV centers of 7 nm. We attribute this finding to a strong quenching of NV centers very close to the surface. These centers consequently do not contribute significantly to the measured PL signal which is dominated by the NVs deeper inside the diamond. Using the distance to the graphene, we estimate a reduction to 75% of the non-quenched PL. We observe a stronger reduction to 50%. However, our simplified consideration does not take into account the unknown absorption of the excitation laser as well as the NV PL due to the graphene agglomerate. We now test if NV ensembles below graphene agglomerates still show the typical spin properties of NV centers. To this end, we record an ODMR of an NV ensemble (see figure 6(d)) below one of the agglomerates, which shows the characteristic ODMR resonance. This result thus strongly indicates that NV centers can serve as multifunctional sensors, which probe the presence of other dipoles as well as the magnetic environment simultaneously.

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NDs are versatile tools in the life sciences as they can enter living cells and organisms. Their applications do not only cover sensing and imaging but also extend to drug delivery and tissue engineering. For recent reviews on these applications see [145–152]. Color centers in NDs first serve as non-bleaching fluorophores and additionally sense temperatures and magnetic fields inside cells. Moreover, novel approaches using the NV center's charge state as a resource allow to sense the electrostatic environment of the NV center [153, 154] and potential changes as small as 20 mV have been measured using this technique.

Considering electronic devices, NDs with NV ensembles enable the simultaneous measurement of local temperature and electric current distribution [155]. These quantities are relevant in the context of high electron mobility electronics, where local heating occurs and potentially induces device degradation (see figure 7(a)). Both parameters can be measured in parallel as the local temperature affects the NV's zero-field-splitting, while the local current density influences the splitting of the m_s = −1 and m_s = 1 states as a result of the local magnetic field. Despite a certain scatter in the zero-field-splitting of individual NDs (see figure 7(b)), the relevant temperature increases can be measured. For this applications, the ability to deposit NDs on almost all desired substrates and devices is highly advantageous.

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Nanostructured SCD especially in the form of scanning diamond pillars has been used to image magnetic features with nanoscale resolution revealing magnetic features like superconducting vortices [133], skyrmions [156, 157] and magnetic properties in antiferromagnets [158]. Moreover, scanning NVs locally imaged the magnetic field arising due to microwave currents in a stripline [131]. As a novel field of application, NV centers in scanning tips have been used to probe the local conductivity of metallic structures [103] and have proven to be useful for the investigation of the magnetic field of a hard-disk write head [159]. To avoid the need for scanning pillars, also the sample under investigation e.g. a cell can be scanned over a stationary pillar. Using this approach, clusters of iron-storage molecules (Ferritin) have been imaged in a single cell [160].

Recently, the discovery of ferromagnetic behavior in two-dimensional van der Waals materials has triggered significant research interest. An example for such a material is CrI₃. Using scanning NV magnetometry (see figure 8(a)), fundamental questions on the magnetism in CrI₃ have been addressed [161]: the measurements show that flakes with an even number of layers show no resulting magnetization in contrast to flakes with an odd number of layers (see figures 8(b)–(d)). Due to the high spatial resolution and the low back-action in this sensing approach, it is possible to observe direct connections between structural and nanoscale magnetic properties of the material. Consequently, magnetic imaging using a scanning single NV center can help to understand magnetic two-dimensional materials which are candidates for future spintronic devices.

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For sensing and imaging approaches using the dipolar nature of NV centers, it is significant to consider the role of the two electric transition dipoles, $\vec{X}$ and $\vec{Y}$. The detection of near-field processes typically involves measuring the lifetime of the interacting systems. The following calculation illustrates how the observed lifetime is influenced by the existence of two dipoles $\vec{X}$ and $\vec{Y}$.

$\vec{X}$ and $\vec{Y}$ are orthogonal to each other and located in the plane perpendicular to the NV axis (see section 3.1, [89, 90]), while local strain determines the in plane orientation $\vec{X}$ and $\vec{Y}$. Without local strain and the NV axis along [111], $\vec{X}$ and $\vec{Y}$ orient as $X\parallel [\bar{1}\bar{1}2]$ and $Y\parallel [1\bar{1}0]$ [90].

Following a semiclassical approximation, the radiative decay rate Γ^rad associated to a transition from an excited state to a ground state is given by [89]

$$\begin{eqnarray}&&{{\boldsymbol{\Gamma }}}^{{rad}}=\displaystyle \frac{{ \mathcal P }}{{\hslash }\omega }.\end{eqnarray} \tag{ 2 }$$

Here, ${ \mathcal P }$ is the power dissipated by the transition dipole $\vec{\mu }$. The lifetime τ for this transition is given as the inverse of the radiative decay rate Γ^rad (assuming a quantum efficiency of unity).

In a homogeneous medium with a dielectric constant ₁, we can write:

$$\begin{eqnarray}&&{{\boldsymbol{\Gamma }}}_{{\hom }}^{{rad}}=\displaystyle \frac{{{ \mathcal P }}_{{\hom }}}{{\hslash }\omega },\qquad {{ \mathcal P }}_{{\hom }}({\vec{\mu }}^{2})={\vec{\mu }}^{2}\sqrt{{\epsilon }_{1}}{\omega }^{4}{/3c}^{3}.\end{eqnarray} \tag{ 3 }$$

Assuming a homogenous dielectric is suitable for an NV center buried deeply in diamond. In this case, the observed decay rate (and hence the lifetime) will be given by the average of ${{\boldsymbol{\Gamma }}}_{{\hom }}^{\vec{X},{rad}}$ and ${{\boldsymbol{\Gamma }}}_{{\hom }}^{\vec{Y},{rad}}$ :

$$\begin{eqnarray}&&{{\boldsymbol{\Gamma }}}_{{\hom }}^{{rad}}=\displaystyle \frac{1}{2}({{\boldsymbol{\Gamma }}}_{{\hom }}^{\vec{X},{rad}}+{{\boldsymbol{\Gamma }}}_{{\hom }}^{\vec{Y},{rad}}).\end{eqnarray} \tag{ 4 }$$

To model a NV center close to the diamond surface, we have to use a a layered system, with a NV center buried at a depth h. We decompose the power dissipated by the two dipoles ${{ \mathcal P }}_{{lay}}^{\vec{X}}$, ${{ \mathcal P }}_{{lay}}^{\vec{Y}}$ in components parallel and orthogonal to the surface:

$$\begin{eqnarray}&&{{ \mathcal P }}_{{lay}}^{\vec{X}}=\displaystyle \frac{{\vec{\mu }}_{\perp ,\vec{X}}^{2}}{{\vec{\mu }}_{\vec{X}}^{2}}{{ \mathcal P }}_{{lay},\perp }+\displaystyle \frac{{\vec{\mu }}_{\parallel ,\vec{X}}^{2}}{{\vec{\mu }}_{\vec{X}}^{2}}{{ \mathcal P }}_{{lay},\parallel }\mathop{=}\limits^{!}{{a}_{\perp }^{\vec{X}}}^{2}{{ \mathcal P }}_{{lay},\perp }+{{a}_{\parallel }^{\vec{X}}}^{2}{{ \mathcal P }}_{{lay},\parallel },\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{{ \mathcal P }}_{{lay}}^{\vec{Y}}=\displaystyle \frac{{\vec{\mu }}_{\perp ,\vec{Y}}^{2}}{{\vec{\mu }}_{\vec{Y}}^{2}}{{ \mathcal P }}_{{lay},\perp }+\displaystyle \frac{{\vec{\mu }}_{\parallel ,\vec{Y}}^{2}}{{\vec{\mu }}_{\vec{Y}}^{2}}{{ \mathcal P }}_{{lay},\parallel }\mathop{=}\limits^{!}{{a}_{\perp }^{\vec{Y}}}^{2}{{ \mathcal P }}_{{lay},\perp }+{{a}_{\parallel }^{\vec{Y}}}^{2}{{ \mathcal P }}_{{lay},\parallel },\end{eqnarray} \tag{ 6 }$$

where ${{ \mathcal P }}_{{lay},\perp }$(${{ \mathcal P }}_{{lay},\parallel }$) is the power dissipated by the dipole component orthogonal (parallel) to the surface. We note that ${{ \mathcal P }}_{{lay},\perp }$ and ${{ \mathcal P }}_{{lay},\parallel }$ as well as all deduced quantities depend on the depth h of the NV center. Equations defining quantities and coefficient are highlighted using $\mathop{=}\limits^{!}$.

Considering equations (2), (3), (5), we find the decay rate ${{\boldsymbol{\Gamma }}}_{{lay}}^{\vec{X},{rad}}$ for $\vec{X}$:

$$\begin{eqnarray}&&{{\boldsymbol{\Gamma }}}_{{lay}}^{\vec{X},{rad}}=\displaystyle \frac{{{ \mathcal P }}_{{lay}}^{\vec{X}}}{{\hslash }\omega }\mathop{=}\limits^{!}{F}_{\vec{X}}{{\boldsymbol{\Gamma }}}_{{\hom }}^{{rad}}=({{a}_{\perp }^{\vec{X}}}^{2}{F}_{\perp }+{{a}_{\parallel }^{\vec{X}}}^{2}{F}_{\parallel }){{\boldsymbol{\Gamma }}}_{{\hom }}^{{rad}},\end{eqnarray} \tag{ 7 }$$

with:

$$\begin{eqnarray}&&{F}_{\perp }\mathop{=}\limits^{!}\displaystyle \frac{{{ \mathcal P }}_{{lay},\perp }}{{{ \mathcal P }}_{{\hom }}},\,{F}_{\parallel }\mathop{=}\limits^{!}\displaystyle \frac{{{ \mathcal P }}_{{lay},\parallel }}{{{ \mathcal P }}_{{\hom }}}.\end{eqnarray} \tag{ 8 }$$

Similarly for $\vec{Y}$

$$\begin{eqnarray}&&{{\boldsymbol{\Gamma }}}_{{lay}}^{\vec{Y},{rad}}=\displaystyle \frac{{{ \mathcal P }}_{{lay}}^{\vec{Y}}}{{\hslash }\omega }\mathop{=}\limits^{!}{F}_{\vec{Y}}{{\boldsymbol{\Gamma }}}_{{\hom }}^{{rad}}=({{a}_{\perp }^{\vec{Y}}}^{2}{F}_{\perp }+{{a}_{\parallel }^{\vec{Y}}}^{2}{F}_{\parallel }){{\boldsymbol{\Gamma }}}_{{\hom }}^{{rad}}.\end{eqnarray} \tag{ 9 }$$

To obtain the total decay rate, we use equation (4):

$$\begin{eqnarray}&&{{\boldsymbol{\Gamma }}}_{{lay}}^{{rad}}=\displaystyle \frac{1}{2}({{\boldsymbol{\Gamma }}}_{{lay}}^{\vec{X},{rad}}+{{\boldsymbol{\Gamma }}}_{{lay}}^{\vec{Y},{rad}}).\end{eqnarray} \tag{ 10 }$$

Considering equations (7) and (9),

$$\begin{eqnarray}\begin{array}{rcl}{{\boldsymbol{\Gamma }}}_{{lay}}^{{rad}} & = & \displaystyle \frac{1}{2}{{\boldsymbol{\Gamma }}}_{{\hom }}^{{rad}}[{F}_{\perp }({{a}_{\perp }^{\vec{X}}}^{2}+{{a}_{\perp }^{\vec{Y}}}^{2})+{F}_{\parallel }({{a}_{\parallel }^{\vec{X}}}^{2}+{{a}_{\parallel }^{\vec{Y}}}^{2})]={{\boldsymbol{\Gamma }}}_{{\hom }}^{{rad}}[{a}_{\perp }^{2}{F}_{\perp }+{a}_{\parallel }^{2}{F}_{\parallel }]\\ & = & \ {{\boldsymbol{\Gamma }}}_{{\hom }}^{{rad}}{F}_{{lay}},\end{array}\end{eqnarray} \tag{ 11 }$$

with:

$$\begin{eqnarray}&&{F}_{{lay}}\mathop{=}\limits^{!}{a}_{\perp }^{2}{F}_{\perp }+{a}_{\parallel }^{2}{F}_{\parallel },\end{eqnarray} \tag{ 12 }$$

and

$$\begin{eqnarray}&&{a}_{\perp }^{2}\mathop{=}\limits^{!}\displaystyle \frac{{{a}_{\perp }^{\vec{X}}}^{2}+{{a}_{\perp }^{\vec{Y}}}^{2}}{2},\,{a}_{\parallel }^{2}\mathop{=}\limits^{!}\displaystyle \frac{{{a}_{\parallel }^{\vec{X}}}^{2}+{{a}_{\parallel }^{\vec{Y}}}^{2}}{2}.\end{eqnarray} \tag{ 13 }$$

Considering in detail the possible orientations of $\vec{X}$ and $\vec{Y}$ in (100)-oriented SCD which is the most common SCD orientation, we find that that a_⊥² and a_∥² do not depend on the orientation of $\vec{X}$ and $\vec{Y}$ in the plane orthogonal to the NV axis, nor on which of the equivalent directions the NV occupies and ${a}_{\perp }^{2}=\tfrac{1}{3}$ and ${a}_{\parallel }^{2}=\tfrac{2}{3}$.

Consequently, Γ^rad_lay is the same for all equivalent orientations of NV axis and orientations of $\vec{X}$ and $\vec{Y}$. Γ^rad_lay and the lifetime of the NV will depend only on the NV depth h. FRET will add a non-radiative channel to the system. In this case the total decay rate Γ_lay^tot for the layered system is:

$$\begin{eqnarray}&&{{\boldsymbol{\Gamma }}}_{{lay}}^{{tot}}={{\boldsymbol{\Gamma }}}_{{lay}}^{{rad}}+{{\boldsymbol{\Gamma }}}_{{lay}}^{{nr}}={{\boldsymbol{\Gamma }}}_{{\hom }}^{{rad}}{F}_{{lay}}+{{\boldsymbol{\Gamma }}}_{{lay}}^{{nr}}.\end{eqnarray} \tag{ 14 }$$

Consequently, we find a unique decay rate ${{\boldsymbol{\Gamma }}}_{{lay}}^{{tot}}$ for an ensemble of NV centers at the same depth h. Hence, the observed PL decay will be monoexponential characterized by a lifetime:

$$\begin{eqnarray}&&\tau =\displaystyle \frac{1}{{{\boldsymbol{\Gamma }}}_{{lay}}^{{tot}}}.\end{eqnarray} \tag{ 15 }$$

Consequently, shalllow NV centers in (100) oriented diamond may form highly-suitable sensors for FRET, despite their non-trivial transition dipoles.